PRL 95, 037201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending15 JULY 2005

Quasi-Two-Dimensional Diluted Magnetic Semiconductor Systems

D. J. Priour, Jr., E. H. Hwang, and S. Das SarmaCondensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

(Received 7 January 2005; revised manuscript received 16 May 2005; published 12 July 2005)

0031-9007=

We develop a theory for two-dimensional diluted magnetic semiconductor systems (e.g., Ga1�xMnxAslayers) where the itinerant carriers mediating the ferromagnetic interaction between the impurity localmoments, as well as the local moments themselves, are confined in a two-dimensional layer. The theoryincludes exact spatial disorder effects associated with the random local moment positions within adisordered RKKY lattice field theory description. We predict the ferromagnetic transition temperature(Tc) as well as the nature of the spontaneous magnetization. The theory includes disorder and finite carriermean-free path effects as well as the important correction arising from the finite temperature RKKYinteraction, finding a strong density dependence of Tc in contrast to the simple virtual crystalapproximation.

DOI: 10.1103/PhysRevLett.95.037201 PACS numbers: 75.50.Pp, 75.10.Nr, 75.30.Hx

Many projected applications of diluted magnetic semi-conductors (DMS), i.e., systems that combine the advan-tages of a ferromagnetic material with those of asemiconductor with the additional flexibility of carrier-mediated ferromagnetism enabling the tuning of the mag-netic properties by applying external gate voltages oroptical pulses to control the carrier density, would involvethe use of two-dimensional (2D) DMS structures such asquantum wells, multilayers, superlattices, or heterostruc-tures. Such 2D DMS structures are also of intrinsic funda-mental interest since magnetic properties in twodimensions are expected to be substantially differentfrom the three-dimensional (3D) systems [1] that havemostly been theoretically studied in the DMS literature.The 2D DMS systems introduce the possibility of gating,and thereby controlling both electrical and magnetic prop-erties by tuning the carrier density. In fact, such a carrierdensity modulation of DMS properties has already beendemonstrated [2,3] in gated DMS field effect heterostruc-tures. For various future spintronic applications, the devel-opment of such 2D DMS structures is obviously of greatimportance.

In this Letter, we provide the basic theoretical pictureunderlying 2D DMS ferromagnetism focusing on the well-studied Ga1�xMnxAs, with x � 0:03–0:08, a system wherethe ferromagnetism is well established to be arising fromthe alignment (for T < Tc � 100–200 K) of Mn local mo-ments through the indirect exchange interaction carried byitinerant holes in the GaAs valence (or impurity) band (thatare also contributed by the Mn atoms which serve the dualpurpose of being the impurity local moments as well as thedopants). Our theory is quite general and should apply toother ‘‘metallic’’ DMS materials where the ferromagneticinteraction between the impurity local moments is medi-ated by itinerant carriers (electrons or holes). One of ourimportant findings is that the ferromagnetic transition(‘‘Curie’’) temperature for the 2D DMS systems typicallytends to be substantially less than the corresponding 3D

05=95(3)=037201(4)$23.00 03720

case with equivalent system parameters. In particular, Tc inthe 2D case is found to be comparable (Tc � TF) to theFermi temperature (TF � EF=kB, where EF is the 2D Fermienergy) of the 2D hole systems. This necessitates that thefull finite temperature form for the 2D RKKY interactionbe used in calculating DMS magnetic properties, furthersuppressing Tc in the system. In fact, this general loweringof the 2D DMS Tc compared with the corresponding 3Dcase is our central new theoretical result. This implies thatspintronic applications involving 2D DMS heterostructureswill be problematic since the typical Tc (at least for thecurrently existing DMS materials) is likely to be far belowroom temperature (Tc < 100 K). A related equally impor-tant theoretical finding is that, although the continuumvirtual crystal approximation (VCA) much used for 3DGa1�xMnxAs physics [4] predicts that the Tc in 2D DMSsystems (being proportional to the 2D density of states) isindependent of 2D carrier density, there is an intrinsiccarrier density dependence of Tc even in the strict 2Dsystem arising from the density and temperature depen-dence of the finite temperature effective RKKY interaction.

Although the Hohenberg-Mermin-Wagner (HMW)theorem precludes long range order in 2D systems withHeisenberg spins, the theorem applies only for the case inwhich the coupling between spins is absolutely isotropic.As has been shown both formally and numerically [5–7],even a small amount of anisotropy is sufficient to stabilizelong range order at finite temperatures. We examine theimpact of anisotropy explicitly by studying the classicalanisotropic 2D Heisenberg model H � �J0

Phiji�S

xi S

xj �

Syi Syj � �� 1SziS

zj�, where J0 is the ferromagnetic ex-

change constant, � is the anisotropy parameter, and thespins occupy a 2D square lattice; the sum is over nearestneighbors only. Using a variant of the Wolff clusterMonte Carlo technique [8], we calculate Tc by findingthe intersection of Binder cumulants [9] for two differentsystem sizes (L � 40 and 80). The results are shown in

1-1 2005 The American Physical Society

0 2 4 6 8 10k r

−1

0

1

2

3

4

5

χ (k

r,T

)

F

F 0 0.5 1 1.5 2 2.5T/T

0

0.2

0.4

0.6

0.8

1

J

/J

F

eff

o

T/T =0.0F

F

F

T/T =0.5

T/T =1.0

FIG. 2. Temperature dependent 2D RKKY range function�kFr; T=TF as a function of kFr for several temperaturesT=TF � 0:0, 0.5, and 1.0. The inset portrays the effective finitetemperature coupling Jeff .

−1 −.75 −.5 −.25 0 .25 .5 .75 10

.25

.5

.75

1

1.25

1.5

1.75

(JT

γ

−.05 −.025 0 .025 .05

.2

.4

.6

γ

)Bk

C0/

C0/

)Bk

T(J

FIG. 1. Curie Temperatures plotted versus �, the anisotropyparameter. The inset displays Tc values for a much smaller rangeof anisotropies. In both images, the Monte Carlo errors aresmaller than the graph symbols.

PRL 95, 037201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending15 JULY 2005

Fig. 1, where it is readily evident that even a small amountof anisotropy yields Curie temperatures in the vicinity ofJ0=kB, a value in reasonable agreement with TMFT

c �43 J0=kB, the corresponding mean field result for the nearestneighbor 2D Heisenberg model. Previous theoretical stud-ies also examined ferromagnetism in the 2D DMS context[10,11], without, however, considering the HMW theoremor the anisotropy issue.

We apply the lattice mean-field theory (MFT) developedpreviously [12], but we also find intriguing results in thecontext of simple continuum Weiss MFT, which demon-strate that it is essential to incorporate finite temperatureeffects in the effective carrier-mediated interaction be-tween Mn impurity moments. We assume the 2D holegas to be confined in the same plane as the Mn dop-ants—it is straightforward to consider [10,11] a spatialseparation between the dopants and the holes as well as toconsider the quasi-2D confinement for the holes. Theseadditional complications would lower Tc below the strict2D limit considered in our model.

Our theory is constructed for two-dimensional DMSsystems in the metallic limit with itinerant carriers (weassume the carrier-mediated effective Mn-Mn indirect ex-change interaction to be of the RKKY form). However,including a finite carrier mean-free path in our theoryallows us to take into account the dependence of themagnetic behavior of our system on the carrier transportproperties. In fact, using an exponential cutoff in the rangeof the RKKY function permits us to treat the long and shortrange magnetic interaction regimes simply by varying thecutoff parameter l. In our system, salient parameters in-clude the Mn local moment concentration (x), the freecarrier density (nc), and the exponential cutoff scale lassociated with the carrier mean-free path. The 2D Fermi

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temperature TF �@2k2F

2m�kBdepends on nc since the 2D kF /

n1=2c , where kF is the Fermi wave vector.Our effective Hamiltonian describes the Mn-Mn mag-

netic interaction between classical Heisenberg spins Si ona 2D lattice: H � �ijJ

RKKYij Si � Sj, where Si is the ith Mn

local moment of spin 5=2. In our lattice MFT we assumethe Mn dopants to lie entirely on the [100] plane of theGaAs zinc-blende crystal lattice, with the impurities occu-pying the Ga sites, which form a square lattice with latticeconstant a. The carrier-mediated RKKY indirect exchangeinteraction describes the effective magnetic interactionbetween Mn local moments induced by the free carrierspin polarization. JRKKYij T � J 0�kFr; T=TF, where�kFr; T=TF is the temperature dependent range functionthat is obtained from the 2D spin susceptibility, and J 0 �

�Jpda �

2 m�

8��@�2a2. We use the parameters Jpd � 0:15 eV nm3

[13], a � 0:4 nm, and m� � 0:4me throughout this work.For T � 0 the 2D RKKY interaction is known exactly

[14]: JRKKYij x � J 0kFa2�J0xN0x � J1xN1x�,where Jnx are the Bessel functions of the first kind, andNnx are Bessel functions of the second kind. At finitetemperatures, it is not possible to obtain the RKKY rangefunction analytically. Thus, we calculate the finite tem-perature RKKY interaction numerically. In Fig. 2, we showthe temperature dependent range function �kFr; T=TF asa function of kFr for various T=TF values. One sees in-creasingly severe thermal damping of the RKKY oscilla-tions with increasing T=TF. The inset of Fig. 2 displays theeffective coupling constant, given by Jeff �

RJRKKYij rdr.

One finds, as expected, that Jeff decreases with increasingtemperature due to the damping of the range function.Since the magnetic properties of the 2D system are directly

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PRL 95, 037201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending15 JULY 2005

dependent on Jeff , a finite T=TF can have an importantimpact.

Via lattice MFT [12], we calculate Tc for the squarelattice with constant a;

Tc �35

12kBxX1

i�1

NiJri; Tc; (1)

where Ni and ri are the numbers and distances of the ithnearest neighbors, respectively. The continuum limit,which we examine first, is attained for l; k�1

F � a, wherea is the lattice spacing. We examine the large l limit (l �k�1F ) and find that Eq. (1) becomes T�

c �T1cR10 �y; T�

c=TFydy � T1c gT�

c=TF, where T�c is the

Curie temperature in the continuum limit, T1c � TcTF !

1 � 35�xJ 0=6kB, and gT�c=TF depends only on the

ratio of T�c to the Fermi temperature TF. For T�

c � TF,gT�

c=TF ! 1, the dependence on carrier concentration islost, and T�

c depends only on the impurity concentration x.This TF ! 1 limit of the continuum MFT is the actualVCA limit. This result, with no dependence on nc, isproportional to the density of states at the Fermi level EF

in two dimensions and reflects the peculiarity of the 2Ddensity of states being independent of carrier density. (Theanalogous continuum 3D result, also proportional to thedensity of states at EF, is T�

c / xn1=3c and does depend onthe carrier density.)

Returning to the lattice MFT, which explicitly takes intoaccount the discreteness of the square lattice, we alsoincorporate the effects of the finite carrier mean-free pathl by including an exponential cutoff in the range of theRKKY interaction. Tc curves are shown in Fig. 3 forseveral Mn dopant concentrations. A salient feature ofthe curves is a marked dependence on nc. The Curie

0 2 4 6 8 100

10

20

30

40

0 2 4 6 8 100

50

100

150

200

x=0.01

x=0.05

x=0.10

x=0.01

x=0.05

T

(K)

c

T

(K)

c

x=0.10

11n (10 cm )−2c

−2n (10 cm )c13

FIG. 3. Lattice MFT 2D Curie temperature curves for variousvalues of x. In the inset, Tc curves are shown for a much greaterrange of carrier density nc. In both the main graph and the inset,l=a � 5.

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temperature increases monotonically in nc over the experi-mentally accessible range of carrier densities. However, forconsiderably higher nc (i.e., approaching 1014 cm�2), non-monotonic behavior is seen in Tc. This is evident in theinset of Fig. 3, where the Curie temperature curves seem toachieve saturation for intermediate carrier densities andultimately begin to decrease as the length scale k�1

F ofthe RKKY oscillations shrinks relative to the lattice con-stant a. Eventually, for k�1

F � a, the discrete nature of thelattice sum in Eq. (1) has a strong effect, leading to aconsiderably smaller Tc than the continuum value, T�

c .This result for 2D Tc including disorder and finite tem-perature RKKY effects is one of our main new results.Tc is strongly affected by the finite size of l; in fact, one

sees that for each of the carrier concentrations representedin Fig. 4, Tc initially increases sharply with increasing l,eventually saturating for l=a � 1. It is informative toexamine the regime a � l; k�1

F , where one can operate inthe continuum limit. We introduce � � kFl as an importantdimensionless variable; as will be seen, the ratio T�

c=TF

tends to zero as � becomes small. For Tc � TF and kFr �1, a reasonable approximation for the finite temperatureRKKY range function is JRKKYr; T � JRKKYr; 0�1�a2 2 � a3 3 � b2 2 � b3 3kFr�e�r=l where �T=TF, a2 � 0:2153, a3 � �0:140, b2 � �1:333, andb3 � 0:862. (The ai and bi have been calculated numeri-cally.) Since JRKKYr; 0 varies slowly with kFr for smallkF, we have for small values of this expansion variable

JRKKYr;T�c �J 0k

2F=��"1�"2kFr

2���1�a2 2

�a3 3�b2

2�b3 3kFr�e

�r=l; (2)

where "1 � ��1=2� �� lnkFr=2�, and "2 ���3=16� �=4� 1=4 lnkFr=2�; � � 0:577 22 . . . is theEuler constant. In calculating T�

c (in the continuum case),we assume that although l � k�1

F , l � a. This additional

0 2 4 6 8 100

10

20

30

40n

c=5.0 ×10

10cm

−2

nc=2.5 ×10

11cm

−2

nc=1.0 ×10

12cm

−2

T

(K)

c

l/aFIG. 4. Lattice MFT Curie temperature curves for variousvalues of nc plotted as a function of the mean-free path l=a.In each case the same Mn concentration, x � 0:05, is used.

1-3

0.01 0.02 0.03 0.04−0.9

−0.6

−0.3

0

0.3

0.6

0.9γ

(con

cavi

ty)

nc=5.0 × 10

10 cm

−2

nc=2.5 × 10

11 cm

−2

nc=1.0 × 10

12 cm−2

0 0.2 0.6 1.00

0.2

0.4

0.6

0.8

1.0

T/Tc

M/M

0

x=0.06x=0.013

x=0.01

x

FIG. 5. Concavity curves versus Mn concentration x for sev-eral carrier densities nc with l=a � 2:0. The inset shows repre-sentative examples of the magnetization, MT. All MT curvesare calculated for nc � 2:5� 1011 cm�2.

PRL 95, 037201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending15 JULY 2005

condition allows the replacement of the discrete formulafor the Curie temperature in Eq. (1) with the continuumversion, and one has T�

c � x 35�6

R10 JRKKYr; T�

c rdr.Carrying out the integration and solving for T�

c , we findT�c=T

1c � �2

� f�ln2� �12� � �2�12 ln

2� �

4116�g, where � �

kFl � 1 and only terms up to fourth order in � are shown.T�c=T1

c is strongly suppressed due to localization effects;when kF is small (i.e., in the small nc limit), the RKKYrange function is very extended relative to a and l. As aconsequence, most of the RKKY interaction is truncated bythe exponential cutoff associated with the finite mean-freepath l. This truncation effect is so severe that T�

c is verysmall in comparison with TF; to fourth order in � there areno corrections to arising from the finiteness of T�

c=TF. Notethat in the opposite limit of kFl � 1, which rarely appliesto DMS systems which are at best ‘‘bad metals,’’ one canobtain the simple formula TckFl � 1 � T�

c �1�kFl

�1=�� by simply considering the disorder inducedsuppression of the 2D density of states.

In addition to Tc, one can also use lattice MFT tocalculate the magnetization MT [12]. The magnetizationbehavior is influenced by the Mn impurity concentration aswell as the form of the effective interaction between Mnlocal moments, in principle specified by the carrier mean-free path l and nc. For convenience, we study MT=Tc;using the normalized temperature scale T=Tc allows asystematic comparison of magnetization profiles corre-sponding to different values of the parameters l, nc, andx. An important trait of the magnetization profile is itsdegree of concavity; concavity in MT is a hallmark of aninsulating system, while convex profiles occur well withinthe metallic regime [4]. Linear magnetization curves cor-

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respond to intermediate impurity densities and mean-freepaths. In terms of the magnetization, the concavity � isgiven by � �

Rt2t1 M

00TdT, or the difference in the slopesof MT. The sign of � indicates whether MT is convex(negative �), concave (positive �), or linear (if � � 0). Thetemperatures t1 and t2 are selected to encompass an inter-mediate temperature range, neither too close to Tc nor tozero. A significant feature of the concavity plots shown inFig. 5 is weak dependence of the concavity of the magne-tization profiles on the carrier density; though the threevalues of nc range over 2 orders of magnitude1010–1012 cm�2, the � graphs lie very close to oneanother. One can also see that the parameter range overwhich the MT profile is concave is much more extensivethan predicted by lattice MFT for three-dimensional DMSsystems [12]. This is primarily a consequence of the ge-ometry of the two-dimensional lattice.

In conclusion, we have considered diluted magneticsemiconductors in quasi-two dimensions. We find thatlong range ferromagnetic order can be stabilized by evena small amount of anisotropy invariably present in reality.We have found that, even at the level of continuum MFT,finite temperature effects in the carrier-mediated effectiveinteraction between Mn impurity moments introduce astrong dependence on the density of carriers, where naiveuse of the zero temperature RKKY formula yields a resultindependent of nc. To take into account the discreteness ofthe strong positional disorder of the 2D DMS system, wehave employed a lattice MFT. Our lattice theory alsoprovides a convenient framework for the inclusion ofimportant physics such as the finite mean-free path. Ingeneral, the 2D DMS Tc is strongly suppressed comparedwith the corresponding 3D DMS Tc, which is somewhatdiscouraging for spintronic applications.

This work is supported by U.S.-ONR and DARPA.

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